MATH 415 (Spring 2021)

Spring 2021
MATH 415-500: Modern Algebra I

Time and venue: TR 1:30–2:45 p.m., ZOOM meeting

First day hand-out

Office hours (ZOOM meeting):

MW 4:00–5:00 p.m.
TR 12:15–1:15 p.m.
by appointment

Office hours during the finals (ZOOM meeting):

Friday, April 30, 1:00–3:00 p.m.
Monday, May 3, 3:00–5:00 p.m.
Tuesday, May 4, 3:00–5:00 p.m.
Wednesday, May 5, 3:00–5:00 p.m.
by appointment

Homework assignment #1 (due Monday, February 1)

Homework assignment #2 (due Friday, February 5)

Homework assignment #3 (due Friday, February 12)

Homework assignment #4 (due Friday, February 26)

Exam 1: Thursday, March 11 (Sample problems)

Homework assignment #5 (due Wednesday, March 17)

Homework assignment #6 (due Monday, March 22)

Homework assignment #7 (due Monday, March 29)

Homework assignment #8 (due Monday, April 5)

Exam 2: Thursday, April 8 (Sample problems)

Homework assignment #9 (due Friday, April 16)

Homework assignment #10 (due Friday, April 23)

Homework assignment #11 (due Friday, April 30)

Final exam: Thursday, May 6, 8:00–11:00 a.m. (Sample problems)

(Optional) homework assignment #12 (no due date)

Course outline:

Part I: Basic group theory

Binary operations
Groups
Subgroups, cyclic groups
Groups of permutations
Cosets, Lagrange's theorem

Fraleigh: Chapters I and II

Lecture 1: Preliminaries.

Fraleigh 0

Lecture 2: Binary operations.

Fraleigh 1–2

Lecture 3: Isomorphism of binary structures. Groups.

Fraleigh 3–4

Lecture 4: Groups and semigroups. Subgroups.

Fraleigh 4–5

Lecture 5: Generators of a group. Cyclic groups. Cayley graphs.

Fraleigh 5–7

Lecture 6: Permutations. Cycle decomposition.

Fraleigh 8–9

Lecture 7: Order and sign of a permutation.

Fraleigh 9

Lecture 8: Definition of the determinant. Cosets. Lagrange's theorem.

Fraleigh 6, 9–10

Part II: More advanced group theory

Direct product of groups
Classification of abelian groups
Homomorphisms of groups
Factor groups
Group actions

Fraleigh: Chapters II and III

Lecture 9: Direct product of groups. Factor groups.

Fraleigh 11, 14

Lecture 10: Homomorphisms of groups. Classification of groups.

Fraleigh 11, 13–14

Lecture 11: Classification of groups (continued). Groups of symmetries. Group action on a set.

Fraleigh 8, 12, 15–17

Lecture 12: Review for Exam 1.

Fraleigh 0–17

Part III: Basic theory of rings and fields

Rings and fields
Integral domains
Modular arithmetic
Rings of polynomials
Factorization of polynomials

Fraleigh: Chapter IV

Lecture 13: Rings and fields.

Fraleigh 18–19

Lecture 14: Follow-up on Exam 1. Advanced algebraic structures.

Fraleigh 0–19, 30

Lecture 15: Rings and fields (continued). Field of quotients.

Fraleigh 1, 18–19, 21

Lecture 16: Modular arithmetic.

Fraleigh 19–20

Lecture 17: Rings of polynomials. Division of polynomials.

Fraleigh 22–23

Lecture 18: Factorization of polynomials over a field.

Fraleigh 23

Lecture 19: Review for Exam 2.

Fraleigh 18–23

Part IV: More advanced ring theory

Ideals
Homomorphisms of rings
Factor rings
Prime and maximal ideals
Factorization in general rings

Fraleigh: Chapters IV and V

Lecture 20: Ideals and factor rings.

Fraleigh 26

Lecture 21: Follow-up on Exam 2. Homomorphisms of rings.

Fraleigh 18–23, 26

Lecture 22: Homomorphisms of rings (continued). Prime and maximal ideals.

Fraleigh 26–27

Lecture 23: Factorization in integral domains.

Fraleigh 23, 45–47

Lecture 24: Euclidean algorithm. Chinese remainder theorem.

Fraleigh 18, 27, 45–46

Lecture 25: Review for the final exam.

Fraleigh 0–23, 26–27

Spring 2021 MATH 415-500: Modern Algebra I